Lightness, hyperspace, and lower oscillation bounds

When does a map admit a lower “anti-continuity” bound like for some function and for all ? The answer is easy: must be injective and its inverse must be uniformly continuous. End of story.

But recalling what happened with diameters of connected sets last time, let’s focus on the inequality for connected subsets . If such exists, the map f has the LOB property, for “lower oscillation bound” (oscillation being the diameter of image). The LOB property does not require to be injective. On the real line, satisfies it with : since it simply folds the line, the worst that can happen to the diameter of an interval is to be halved. Similarly, admits a lower oscillation bound . This one decays faster than linear at 0, indicating some amount of squeezing going on. One may check that every polynomial has the LOB property as well.

On the other hand, the exponential function does not have the LOB property, since tends to as . No surprise there; we know from the relation of continuity and uniform continuity that things like that happen on a non-compact domain.

Also, a function that is constant on some nontrivial connected set will obviously fail LOB. In topology, a mapping is called light if the preimage of every point is totally disconnected, which is exactly the same as not being constant on any nontrivial connected set. So, lightness is necessary for LOB, but not sufficient as shows.

Proof. Suppose not. Then there exists and a sequence of connected subsets such that and . We can assume compact, otherwise replace it with its closure which we can because .

The space of nonempty compact subsets of is called the hyperspace of ; when equipped with the Hausdorff metric, it becomes a compact metric space itself. Pass to a convergent subsequence, still denoted . Its limit has diameter at least , because diameter is a continuous function on the hyperspace. Finally, using the uniform continuity of we get , contradicting the lightness of .

A homeomorphism lacking LOB

Here is another example to demonstrate the importance of compactness (not just boundedness) and continuity: on the domain define . This is a homeomorphism, the inverse being . Yet it fails LOB because the image of line segment has diameter , which can be arbitrarily close to 0. So, the lack of compactness hurts. Extending to the closed square in a discontinuous way, say by letting it be the identity map on the boundary, we see that continuity is also needed, although it’s slightly non-intuitive that one needs continuity (essentially an upper oscillation bound) to estimate oscillation from below.

All that said, on a bounded interval of real line we need neither compactness nor continuity.

Theorem 2: If is a bounded interval, then every light map admits a lower oscillation bound.

Proof. Following the proof of Theorem 1, consider a sequence of intervals such that and . There is no loss of generality in considering open intervals, since it can only make the diameter of the image smaller. Also WLOG, suppose and ; this uses the boundedness of . Consider a nontrivial closed interval . For all sufficiently large we have , which implies . Thus is constant on , a contradiction.

The property that distinguishes real line here is that nontrivial connected sets have nonempty interior. The same works on the circle and various tree-like spaces, but fails for spaces that don’t look one-dimensional.